How the independent arguments sum up in our judgement - logarithmic updating and its interpretation

The remark that Bayes factors due to independent pieces of evidence multiply together and the overall factor finally multiplies the initial odds suggests a change of variables in order to play with additive quantities.¹⁴This can be done taking the logarithm of both sides of Eq. (19), that then become

$$\log_{10}[O_{i,j}({\mbox{\boldmath$E$}},I)] = \sum_{k=1}^n \log_{10}[\tilde O_{i,j}(E_k,I)] + \log_{10}[O_{i,j}(I)]\,,$$ (21)

respectively, where the base 10 is chosen for practical convenience because, as we shall discuss later, what substantially matters are powers of ten of the odds.

Introducing the new symbol JL, we can rewrite Eq. (21) as

$$_{i,j}({\mbox{\boldmath$E$}},I) = _{i,j}(I) + \sum_{k=1}^n \Delta _{i,j}(E_k,I)$$ (22)

$$= _{i,j}(I) + \Delta _{i,j}({\mbox{\boldmath$E$}},I)$$ (23)

or

$$\Delta _{i,j}({\mbox{\boldmath$E$}},I) = _{i,j}({\mbox{\boldmath$E$}},I) - \mbox{JL}_{i,j}(I)\,,$$ (24)

where

$$_{i,j}({\mbox{\boldmath$E$}},I) = \log_{10}\left[O_{ij}({\mbox{\boldmath$E$}},I)\right]$$ (25)

$$_{i,j}(I) = \log_{10}\left[O_{i,j}(I) \right]$$ (26)

$$\Delta _{i,j}(E_k,I) = \log_{10}\left[\tilde O_{i,j}(E_k,I)\right]$$ (27)

$$\Delta _{i,j}({\mbox{\boldmath$E$}},I) = \sum_{k=1}^n \Delta _{i,j}(E_k,I)\,.$$ (28)

Figure: Judgement leaning.

The letter `L' in the symbol is to remind logarithm. But it has also the mnemonic meaning of leaning, in the sense of `inclination' or `propension'. The `J' is for judgment. Therefore `JL' stands for judgement leaning, that is an inclination of the judgement, an expression I have taken the liberty to introduce, using words not already engaged in probability and statistics, because in these fields many controversies are due to different meanings attributed to the same word, or expression, by different people (see Appendices B and G for further comments). JL can then be visualized as the indicator of the `justice balance'¹⁵(figure 3), that displays zero if there is no unbalance, but it could move to the positive or the negative side depending on the weight of the several arguments pro and con. The role of the evidence is to vary the JL indicator by quantities $\Delta$JL's equal to base 10 logarithms of the Bayes factors, that have then a meaning of weight of evidence, an expression due to Charles Sanders Peirce [6] (see Appendix E).

But the judgement is rarely initially unbalanced. This the role of JL$_{i,j}(I)$, that can be considered as a a kind of initial weight of evidence due to our prior knowledge about the hypotheses $H_i$ and $H_j$ [and that could even be written as $\Delta$JL$_{i,j}(E_0,I)$, to stress that it is related to a 0-th piece of evidence]

To understand the rationale behind a possible uniform treatment of the prior as it would be a piece of evidence, let us start from a case in which you now absolutely nothing. For example You have to state your beliefs on which of my friends, Dino or Paolo, will first run next Rome marathon. It is absolutely reasonable you assign to the two hypotheses equal probabilities, i.e. $O_{1,2}=1$, or JL$_{1,2}=0$ (your judgement is perfectly balanced). This is because in Your brain these names are only possibly related to Italian males. Nothing more. (But nowadays search engines over the web allow to modify your opinion in minutes.)

As soon as you deal with real hypotheses of your interest, things get quite different. It is in fact very rare the case in which the hypotheses tell you not more than their names. It is enough you think at the hypotheses `rain' or `not rain', the day after you read these lines in the place where you live. In general the information you have in your brain related to the hypotheses of your interest can be considered the initial piece of evidence you have, usually different from that somebody else might have (this the role of $I$ in all our expressions). It follows that prior odds of 10 ( JL$=1$) will influence your leaning towards one hypothesis, exactly like unitary odds ( JL$=0$) followed by a Bayes factor of 10 ( $\Delta$   JL$=1$). This the reason they enter on equal foot when ``balancing arguments'' (to use an expression à la Peirce - see the Appendix E) pro and against hypotheses.

Table: A comparison between probability, odds and judgement leanings

Judg. leaning	Odds(1:2)	$P(H_1)$		Judg. leaning	Odds(1:2)	$P(H_1)$
[ JL$_{1,2}$]	[$O_{1,2}$]	(%)		JL$_{1,2}$]	[$O_{1,2}$]	(%)
0	1.0	50
$-0.1$	0.79	44		$0.1$	1.3	56
$-0.2$	0.63	39		$0.2$	1.6	61
$-0.3$	0.50	33		$0.2$	2.0	67
$-0.4$	0.40	28		$0.4$	2.5	71
$-0.5$	0.32	24		$0.5$	3.2	76
$-0.6$	0.25	20		$0.6$	4.0	80
$-0.7$	0.20	17		$0.7$	5.0	83
$-0.8$	0.16	14		$0.8$	6.3	86
$-0.9$	0.13	11		$0.9$	7.9	89
$-1.0$	0.10	9.1		$1.0$	10	91
$-1.1$	0.079	7.4		$1.1$	13	92.6
$-1.2$	0.063	5.9		$1.2$	16	94.1
$-1.3$	0.050	4.7		$1.0$	20	95.2
$-1.4$	0.040	3.8		$1.4$	25	96.2
$-1.5$	0.032	3.1		$1.5$	32	96.9
$-1.6$	0.025	2.5		$1.6$	40	97.5
$-1.7$	0.020	2.0		$1.7$	50	98.0
$-1.8$	0.016	1.6		$1.8$	63	98.4
$-1.9$	0.013	1.2		$1.9$	80	98.8
$-2.0$	0.010	1.0		$2.0$	100	99.0

Finally, table 1 compares judgements leanings, odds and probabilities, to show that the human sensitivity to belief (that is something like Peirce's intensity of belief - see Appendix E) is not linear with probability. For example, if we assign probabilities of 44%, 50% or 56% to events $E_1$, $E_2$ and $E_3$ we do not expect one of them really more strongly than the others, in the sense that we are not much surprised of any of the three occurs. But the same differences in probability produce quite different sentiment of surprise if we shift the probability scale (if they were, instead, 1%, 7% and 13%, we would be highly surprised if $E_1$ occurs).

Similarly 99.9% probability on $H$ is substantially different from 99.0%, although the difference in probability is `only' 0.9%. This is well understood, and in fact it is known that the best way to express the perception of probability values very close to 1 is to think to the opposite hypothesis $\overline H$, that is 0.1% probable in the first case and 1% probable in the second - we could be quite differently surprised if $H$ does not result to be true in the two cases!¹⁶

From the table we can see that the human resolution is about 1/10 of the JL, although this does not imply that a probability value of 53.85% ( JL$=0.0670$) cannot be stated. It all depends how this value has been evaluated and what is the purpose of it.¹⁷